Which Branch Should I Join as a Civil Engineering graduate (Army, Navy or AF)?

It depends on what you see as your ultimate future. That's when you're done with the MIL stuff. Take a look at the varied progression plans for engineers. If your plan is to be a registered PE, then you want a program that will support it. For the Navy, it's pretty simple. You WILL get your Seabee Warfare Quals done. You WILL go to Grad School, You WILL get your Contracting Warrants. You WILL get your PE ticket. That's if you want to make O-5. Risky going for O-4 as the people who don't have it usually are in the third crunch at the promotion board. The Navy plan is pretty specific and you need to be shrewd in making sure you hit all the pedigree buttons. I'm sure the AF and AR have specific preferred paths. The problem with the AF is much of their construction contracting is done by AR and Navy, so there's experience you'd find hard to get. AR experience that translates best to the outside is COE work. They do facilities, real estate, waterways, dams, etc. Those are different billets vs. demolition, sapper, revetments, etc. which have limited commercial application.

BTW, registration for PE is trending to require more schooling than a BS. Your Grad work needs to be in an engineering discipline that you want to get your ticket in. It was a terrific journey for me in doing real engineering on all the continents (yes, Antarctica too). To do what you're contemplating, you'll have to be evermore moving forward, lest you never get there and become disillusioned. But the journey will be worthwhile. Remember the old Chinese Proverb: "May you live in interesting times."

If you want to actually do engineering the federal service is not for you. There is very, very little actual engineering going on by federal staff, military or civilian. It is much easier and better to have the concept drawings done by in staff personnel, but the actual engineering is done by a contract, awarded to an A&E Firm that specializes in the type of building to be designed. As such, if you really want to design and do engineering commissioning into the reserves or National Guard is the best place for you, with a full time job at a private company.
If you would rather be active duty, I would highly recommend looking at the Navy, you'll get the best duty locations and never need to serve on a ship. Sure beats Minot North Dakota or El Paso Texas.
My background is an Army Engineering Officer who works his civilian job as a Federal Construction Contracting Officer.

Good luck.

PFC (Join to see), I can't comment on the Engineer side of the house and there are some great responses on here...some bad ones too. There is NO guarantee that you will get to be an Engineer if you join. Not in the Army at least. That's if you go through a program like ROTC. Your branch is determined by a board that meets and determines what you will do. Having a Civil Engineering degree would help, but it's no guarantee. You could end up as a Field Artillery Officer. If you can find a direct commission program, that's the only way to guarantee...but those are few and far between. Engineers in the military don't do what you are talking about, even in USACE.