Gravy with Scrambled Eggs :: What went wrong?
Instant gravy packets are not known for their subtle flavor. They contain boullion, tons of salt, and some modified starch. If you don't like the flavor of a lot of that gravy then don't use it.

The scrambled egg recipe that you linked to is of a particular variety. They are just barely set. This gives them a very delicate flavor and even more delicate texture. They are also as moist as scrambled eggs can be.

Such a preparation of eggs would not be well served by a gravy.

My suggestions are these:
1- Use a different recipe for your eggs that will cook them much more. If you insist on a gravy then your eggs will need to have the structure enough to support it.  

2- Make your own gravy and don't ever use those packets again- even if you are starving to death I think they would only hasten your demise. Instead find a recipe for a quick white sauce- (just fat, flour, and milk). If you would like you can add a little shredded cheese for flavor. I think you will find this will be more satisfying and possibly even less expensive than your gravy-packet-crimes-against-humanity.