Yes, you should be able to use the OverDrive app internationally. You will have access to titles you've downloaded in the app, and as long as you have an internet connection, you should be able to borrow and download new titles as usual. However, please keep in mind that some countries block certain types of internet traffic, including library downloads. In this case, OverDrive would have no control over the restriction. These situations are rare.
Your best option may be to borrow and download a title before your international travel to preempt any download issues (or coverage lapses) abroad.