the National Center for Veterans Analysis and Statistics is a good general starting spot for knowing what data is available about veterans. The "reports" page there includes some higher level data about Veteran employment, income, and poverty status. However, this is mostly aggregate and historical data.
Generally, the source for monthly employment information (not limited to veterans) is the Current Population Survey. The CPS makes a special data collection about veterans every August. (Technical documentation for the August 2015 Veterans' Supplement) The main page for data and reports from the Veterans' Supplement is http://www.bls.gov/news.release/vet.toc.htm As far as I can tell, the CPS only publishes annual counts of veteran employment status, and that page only has 2015 data, and for one table 2014/2015 comparable statistics.
Getting access to bulk data from the Current Population Survey is complicated, and not something I'm very expert in. You might be able to get historic data from DataFerrett, but you have to be able to run Java applets in your browser, which is increasingly difficult to do for security reasons. (This answer on the Open Data SE has some more about getting DataFerrett to run.)
It looks like this big, um, colorful page from the National Bureau of Economic Research (NBER) provides direct access to download the data -- you'd be looking at the "August" column.
Another source for historic CPS data is IPUMS -- here's their page about the Veterans supplement but IPUMS is pretty complex also.
If all of this seems harder than it ought to be, I'd agree. You might consider trying a local library, where reference specialists are often available to help with this kind of research. The Census also organizes networks of State Data Centers and Census Information Centers who are often good resources for this kind of thing.