This page features a variety of musical exercises to help oboists master the left and right Eb fingerings.
Notice that the key signature tells you to play all of the B's, E's, A's, and D's as flat. Since Ab is played using the left pinky, you must play the Eb's using the standard right Eb fingering.
Notice that the key signature includes Bb, Eb, and Ab. Since the low C's must be played using the right pinky, you'll need to play the Eb's with your left pinky Eb fingering. As a bonus, you'll need to choose your F fingerings carefully as well. Think about how your ring finger is used in notes immediately preceding and following the F's.
Although this song is relatively easy, the use of our right pinky to play the Db's forces us to use our left Eb fingering. Keep in mind that accidentals carry through to all of the same notes in that measure.
This version of Hot Cross Buns has three flats in the key signature (Bb, Eb, and Ab). Since our right pinky isn't used to play anything other than the Eb key, we can use our standard right Eb fingering for all of these Eb's. Think about what F fingering would be the easiest to use in this situation.
This familiar melody uses Eb's and forked F's. The Eb's could be played in the left hand, however your right pinky will already be on the Eb key as the resonance key for the forked F's, so you might as well play these Eb's with the standard right fingering. Remember to lift the resonance key for the G's.