If it were me, this is what I'd do:
Let's say the sheet with the data you posted above is called "Sheet1"; create another sheet called "Next Month" (or whatever makes sense to you).
In the new sheet, put this formula in A1 (a variation on Nick's formula above):
=ArrayFormula(IF(ROW(Sheet1!B:B)=1, "Date of Service", IF (Sheet1!B:B="","",DATE(YEAR(Sheet1!B:B),MONTH(Sheet1!B:B)+1,DAY(Sheet1!B:B)))))
- Now, each month when you want to replace the dates in Sheet1!B:B, just select 'Next Month'!A:A, copy to clipboard (Ctrl-C), select Sheet1!B:B, and choose Edit > Paste Special > Paste values only.
The old dates and header will be replaced, and the column in 'Next Month' will auto-update again, ready for you to do the same next month, perpetually.