You can use the QueryBuilder class to make the code a bit more readable:
Mongo m = new Mongo();
m.setWriteConcern(WriteConcern.SAFE);
DBCollection c = m.getDB("test").getCollection("or-test");
c.drop();
c.insert(new BasicDBObject("a", "abba"));
c.insert(new BasicDBObject("b", "abba"));
c.insert(new BasicDBObject("a", "bbba"));
c.insert(new BasicDBObject("b", "bbba"));
Pattern pattern = Pattern.compile("^a");
DBObject query = QueryBuilder.start().or(
QueryBuilder.start("a").regex(pattern).get(),
QueryBuilder.start("b").regex(pattern).get()
).get();
System.out.println(c.find(query).count());
This will print "2".