is the data type you're loading into the grid actually using a bit (logical) value for that field, or is it using a text value that you're then trying to map back to a bit value? if your data type is a legitimate logical(true/false) data type, then you should see a checkbox in an editable grid, like so:
i suppose there's also the possibility that your source datatype is a Logical (nullable) value. see the difference here:
here's the data type configuration for that grid:
so i guess my next question is, how is your data type structure being built? is it the result of a sql query component? did someone build the data type directly in an integration library?
if someone built the data type, change the logical (true/false) column to a non-nullable version and you'll get a checkbox instead, making the issue you've described moot.
assuming you already have null values in a database table (assuming also that you're utilizing sql data tables, which, from your description above, sounds like you are), you'll first need to set any null values to 0 (zero) so that your integration components don't error out when they find a null value where they expect a non-nullable true/false value.