OK. Tackle things in order:
Your original question was – "I wonder if anyone knows of any writing from the first century or before that shows a phrase stating a specific number of days and/or a specific number of nights when it absolutely couldn't have included at least a part of each one of the specific number of days and at least a part of each one of the specific number of nights?"
To which the answer is – yes – biblically, the phrase three days and three nights is used idiomatically, as shown by the following:
1 Samuel 30:12 13:
"And they gave him a piece of a cake of figs, and two clusters of raisins: and when he had eaten, his spirit came again to him: for he had eaten no bread, nor drunk any water,
three days and three nights. And David said unto him, To whom belongest thou? and whence art thou? And he said, I am a young man of Egypt, servant to an Amalekite; and my master left me, because
three days agone I fell sick."
In his own words, the young man says he fell ill three days ago. So we may suppose that if today is Thursday, he could well have fallen ill on the Tuesday ... the point here is that the phrase 'three days and three nights' is an emphatic idiom to suggest his poor condition. From his own account, he was ill two days and two nights, and rescued on the third day.
Esther 4:15-16 & 5:1
"Then Esther told them to reply to Mordecai: 'Go, gather all the Jews who are present in Shushan, and fast for me; neither eat nor drink for three days, night or day. My maids and I will fast likewise. And so I will go to the king, which is against the law; and if I perish, I perish!' ... Now it happened on the third day that Esther put on her royal robes and stood in the inner court of the king’s palace, across from the king’s house, while the king sat on his royal throne in the royal house, facing the entrance of the house."
So Esther declares she will not eat nor drink for three days, "night or day", but in fact presents herself to the king on the third day. Again, the "three days and three nights" is used in an idiomatic sense for emphasis.
Coming closer to period, Rabbi Eleazar ben Azariah, who in the 1st century, is quoted as saying:
"A day and night are an Onah ('a portion of time') and the portion of an Onah is as the whole of it"
(
Jerusalem Talmud: Shabbath 9, 3)
This in a discussion of purity laws, Azariah is saying that any portion of a twenty-four hour period could be considered "the whole of it", so your 6th day people are justified in seeing no contradiction, according to idiomatic and legal Hebrew – although I doubt they'd have bothered to find that out.
+++
More interesting is the broader idiom of three days and three nights.
The symbolic meaning of the number of days mentioned in the book of Jonah
In Ancient Near Eastern literature, the phrase "three days and three nights" also appears to be closely associated with death.
In Inanna/Ishtar’s Descent to the Netherworld, she instructs her divine minister, Ninshubur to set up a lament for her if she does not return from the abode of her sister Ereshkigal. When she dies at her sister’s hands, it reads that "After three days (and) three nights had passed, her minister Ninshubur, her minister of favorable words, her knight of true words, sets up a lament for her by the ruins …" (Part II, lines 169–73, cf. Landes 1967:448–449).
It was thus also believed that if someone appeared to be in the ‘realm of death’ for 3 days and three nights that they could only be brought back to life through divine intervention. The realm of death was also called or associated with the grave, the underworld or the depths of the sea ... Also, it appears to have been an ancient belief that when a body did not show signs of life for 3 days, death was considered to be final. This appears to be the case in John 11, where reference is made to the resurrection of Lazarus on the fourth day.
Also, from examples such as the Persian Vendidad, Homer’s Iliad, the New Testament (John 11:17, 11:39) and rabbinical literature, it would appear that ...
"The expression ‘three days and three nights’ is seen to reflect the conception that death is permanent only after a body has shown no signs of animation for a period of three days, the idea being that until that time had elapsed, the soul was conceived as still lingering near the individual, encouraging the hope of revival."
This time span has then often been associated with travel in or to the netherworld by modern commentators.
Of special significance are instances in the Hebrew Bible where it employs the ‘three day’ motif to refer to the length of a journey. In several examples, a trip is completed ‘on the third day’. On other occasions, some kind of travel is indicated as taking place or coming to an end within a span of three days.
A more plausible argument is that a period of 3 days is ‘the absolute limit of human endurance’, with the meaning ‘to the (absolute) limit’ or ‘to the bitter end’. See for instance 2 Samuel 24:11–12 where David chooses 3 days of pestilence as punishment ‘for transgressing the prohibition on the population count’.
Pertaining to the use of the number three, which supports the hypothesis above, is that it can indicate ‘a conventionally complete set’, indicating ‘completeness or full effect’. It can also indicate a ‘considerable lapse of time’ (cf. Ex 1:18; 5:3; 23:17; Is 20:3; 2 Sm 6:11; etc.)
In all likelihood, ‘three days and three nights’ in the context of the book of Jonah refer to the time for a complete act to occur, namely Jonah’s travel in the fish. In all likelihood, it can also be understood that this was the limit of the punishment he could endure before it became too much, evoking the lament that he utters in Jonah 2:3–10. In the light of the mention of the fish’s bowels in parallel to Sheol in Jonah 2 that the fish is simultaneously Jonah’s vehicle of salvation and Sheol, it would appear that this hypothesis for understanding the reference to ‘three days and three nights’ is the most likely one.
Sorry to quote at length, but I thought it really interesting.
What it evidences is not simply that 'three days and three nights' was understood in its idiomatic sense, but also that the idiom referred to much more than simply a measure of time.