From what I have read, you would put two single quotes:

UPDATE Distributor SET last_name = 'GALLANT' ,
first_name = 'DARYL' , middle_name = '' ,
company_name = 'TWO G''S INC.'
WHERE dar = '1261'


You could substitute your AllTrim(Evaluate(field)) With
StrTran(AllTrim(Evaluate(field)),"'","''")

Let me know if this works.