@Elijah: If there is NOTHING (as in all your work and educational records since high school have been changed, and you do not have any criminal record) under your old name how about following my suggestion which I've mentioned in numerous posts here on Susan's: Put down a statement like "None that any relevant records are under" or something similar if you want to "cover" yourself. They'll probably assume you were adopted or something similar (and if the name on your birth certificate has been changed you can say that's the reason for your statement if they try to inquire further). Unfortunately, if ANYTHING they may want to check is under your old name then you'll have to explain (and if you try to omit employers in hope they don't find out that can get you in trouble since people often fudge their work history in hope a new employer doesn't learn about unfavorable prior employment; that's why they often ask you to certify any gaps in employment). Likewise if your educational records are under your old name when they go to check that a message from the school like "New Name never attended here" if they aren't aware of the name change, and they'd assume you were claiming false educational credentials which is another problem in the general workforce. If you don't reveal a name (or alias, so it's not just former "legal" names) that your criminal history is under that is even worse; you could get into trouble for trying to conceal your criminal past.
@ornismom: What I said above should apply to you; if your name was changed prior to your first job and all your relevant school (and other) records are under your new name, you have no more need to explain your old name than someone who was adopted at such age.
Also, to Elijah, do you remember exactly how was the question phrased? If they ask outright if your name was changed, or if you've had any other legal names without further qualifications, then that is legally dicey on their part (for reasons of discrimination issues not confined to the TG community). All they're supposed to inquire are other names (whether or not they were your "legal" name, which could also include pseudonyms known by to your references) which are needed to check relevant records (e.g. work history, school records, criminal background). Remember a name that you went by that was not necessarily your legal name may be relevant, and some former legal names may not be relevant. If your name was changed before you had any such records, or if ALL of them are updated, then there is no need for them to know (but remember if ANY of them are under another name then you will need to mention it).
ETA: I found this post by someone (apparently who had some kind of childhood name change) ask if they needed to mention their original name; the consensus there is they don't if nothing they need to check is under it (#6 is the relevant question):
http://www.askamanager.org/2013/03/short-answer-sunday-7-short-answers-to-7-short-questions-32.htmlETA2: Moderator gave me permission to link directly.