yeah, selecting 'min' as well as some other data point may not get you what you want. Chances are you'll get the carrier value of the first record in the table with customer_id=11554 and then min_two_day would actually be the lowest value of two_day which could be an entirely different record. You could try something like this:
SELECT carrier, two_day AS min_two_day FROM quotes WHERE customer_id='11111' ORDER BY two_day ASC LIMIT 1
That picks the one record with customer_id=11111 that has the lowest value for two_day and returns its carrier value. Note that if you had two different carriers with the same two_day value, it would just pick the first one in the database. This could result in favoring one particular record.