Can I find out why an aircraft squawked an emergency code (7x00)?
This FAA webpage has a list of the various transponder codes and what they are used for, which will give you an idea of what the problem is.  Unless there is an actual aircraft incident or accident which requires the FAA to investigate though, you won't be able to find a record of it.  In fact, pilots aren't even required to file a report unless requested, and those don't go into a public database.

For actual incidents or accidents, you can look at the NTSB Accident Database &amp; Synopses.

For pilot reports to the NASA ASRS self-reporting safety system, you can search their database.  If the pilot filed a report then you will see it show up there.

Commonly used emergency codes are:


7500 - Hijack Code
7600 - Loss of Radio Communications Code
7700 - Emergency Code


Also, some countries use the code 7000 for VFR traffic (which is not an emergency code), but the US uses 1200 instead.