Storing .h files in SQLite using SQLJet
Consider in the following:

public Database(String dbName) {
    this.dbName = dbName;
}



I would check for a null dbName being passed in. Note that you pass this reference and then simply store it. Consequently, if it's null, you won't find out until later (perhaps, much later). You then have to work out at what point that was set to null.
Will you change dbName ? If not, make it final. It'll stop you changing it inadvertently later on. Immutability is often a good idea. It makes the class more robust and thread-safety easier to achieve. Perhaps not a requirement here, but who knows ? It's easier to relax a restriction rather than apply it after-the-fact.