Do you mean that you need to omit all of the records shown in red in the following example?
Field A Field B
Apple Red Apple
In other words, the exact text contained by Field B can be different on each record depending on what value is entered into A?
If so, I'd set up a calculation field:
Patterncount ( Field B ; Field A )
And specify ">0" in the calculation as my criteria to find or omit records
Field A Field B June 2014
Field A changes monthly. When it does, the contents get pasted into Field B, adding to what's already there. So, I only want to Omit records where Field B contains "June 2014" or whatever is in the current Field A
That matches the assumptions I used in posting my previous answer. What I suggested there should work for you.
the contents get pasted into Field B, adding to what's already there
Does not sound like the optimum design to use for this in your database. Setting up a related table with one related record for each date can be much more flexible a way to deal with such a set of dates.(And if you are using copy and paste script steps to put these dates into this field, that is also not the best way to do this.)
Correction, That doesn't match my original assumptions.
In a scripted find,
Set Variable [$Date ; YourTable::Field A ]
Enter find mode 
Set field [ YourTable::Field B ; "*" & $Date & "*" ]
will omit all records where the date in Field A of the current record is found anywhere in the list of dates in Field A.
My comments about less than optimum design choices still apply.
Oh, yes, its not great design (was "inherited" that way and I haven't gotten around to changing it since its only used in this one instance).
Ok, I got it to work! Left my working script as it was and just added the Set Variable part before my Constrain Found Set (which omits records where Field B: [$$variable]
Note: a self join relationship between Field A of your current layout's table occurrence and Field B of a second occurrence of the same table and using the ≠ operator instead of the = operator would match to all records that do not have this month, year text listed in Field B.
YourTable::Field A ≠ YourTable 2::Field B
I don't know how to use a (not equal to) symbol in the Find Requests, since is not an operator (why not??). So, I'm not sure how to do that within the same script.
I haven't suggested that you use the ≠ in a find request. I've posted that you can use that operator in a relationship that you'd set up in Manage | Database | Relationships. "table occurrences" are the "boxes" found in that window that you use to define relationships in your database. You can create multiple Tutorial: What are Table Occurrences? that all refer to the same data source table.