OK-- so far I have…
The Third of June from “Ode to Billy Joe”
The Third of September from “Papa was a Rollin Stone”
That only leaves 363 dates to go! Can you help me round out the calendar?
“Early morning, April 4…” U2 - Pride (In The Name Of Love)
Sublime - April 29, 1992 (Miami)
“Was it here or yesterday, or wasn’t it the 14th of November?” - “Things In My Life” by Men Without Hats
“May 1, 1990”: from the song of that name by Adrian Belew
“I think it was the fourth of July”
Saturday in the Park – Chicago
But the date in the lyrics is April 26, 1992! The OP asks for lyrics 
4th of July - ‘Saturday in the Park’ by Chicago
I’ll be darned, you’re right..I never caught that.
Morphine - “Radar”
If I am guilty so are you
It was March 4th, 1982
“Do you remember/ The 21st night of September?” - September, ** Earth, Wind & Fire**
On August 1st, 1981… I traveled to Scott’s house with a BB Gun. - Barenaked Ladies “I Live With it Everyday”
Also “Fourth of July” originally written and recorded by Dave Alvin but also recorded by X, which is the version I first heard.
“Remember the fifth of November.” – John Lennon “Remember”
Would this count?
All is quiet on New Year’s Day - “New Years Day” - U2
“It’s December the 24th…” The Carpenter’s “White Christmas”
Alas already we have a duplicate as that date is also the subject of Neil Diamond’s Desiree. However Desiree moves on to the Fourth of June so we’ll put it down for the latter.
Later the same week, we have the dark of the moon in the sixth of June during Convoy.
“The first of December was covered with snow.” Sweet Baby James, James Taylor
“I married Isis on the 5th day of May,
But I could not hold on to her very long.”
-Bob Dylan, “Isis,” from the brilliant*** Desire ***album
Good one!
The Rainmakers - “One More Summer”
My genie got married to a CPA
April 15th is her wedding day
“Now it’s boilin’ in a miserable March 21st.” - Tom Waits, “Town with No Cheer”
Al Stewart: “The Last Day of June, 1934”