You could also try:
Formula: copy to clipboard
=IF(MONTH(EOMONTH(A1,0)+1)=MONTH(A10),"Anniv Month","")

Don't know if this will help with your data layout, but to show the month of the anniversary in a text format:
Formula: copy to clipboard
=TEXT(EOMONTH(A1,0)+1,"Mmmm")
This will return 'January' if the month of hire was December, for example.