In TSQL, how to evaluate an expression and assign it to a BIT field?

You can do this with a CASE statement:

DECLARE @bitvar BIT 
DECLARE @search_term varchar(128)

set @search_term = 'abc'

SET @bitvar = CASE 
                 WHEN (@search_term = 'abc') THEN 1
                 ELSE 0
              END

select @bitvar