Here's how to do the first part:
=IF(COUNTIF(B1:H1,A1)=0,"Unique", "Not unique")

There's a logic problem in the second part of your question: "to tell
me if the date in cell B1 is unique in comparison to all the other
dates in the range A1 through H1". B1 is included in that range. Or
do you need to check if B1 is repeated in A1 and C1:H1?