Nikon 55-300 mm DX vs FX?

I have nikon d7000. I know that FX lenses are used in larger sensor size. Still if I use nikon 55-300 mm FX lens rather than 55-300 DX lens, will I get better picture?

The 55-300mm IS a DX lens - if you use it on a full-format body, the image will be cropped to DX size.

Picture quality doesn't come into it, no matter which you use, but some FX lenses can't autofocus on the D3000/D5000 series due to their lack of body motors - not a problem with the D7000.

If you mean the more expensive 70-300mm AF-S VR (not the cheap 70-300mm), the good 70-300mm is reputed to be a little sharper than the 55-300mm, with better edge sharpness according to SLR Gear. It is also reputed to autofocus faster and sometimes more accurately than the 55-300mm according to Mansurov.
The fact that the 70-300mm is an FX lens does not prevent you from using it on a DX body.

To get best results with any lens, and especially with telephotos, it helps to use a tripod. (Turn off VR when using a tripod.) VR can help a little handheld, but not as much as using a tripod. The 55-300mm gets kind of front heavy, and the 70-300mm more so. Heavier lenses than the 70-300mm usually have a tripod collar to move the balance point forward. A good, hefty tripod head helps. Many lighter duty heads will tend to sag a little between your supporting it to aim and getting it locked down. The Manfrotto 460MG 3D Magnesium Head with RC2 Quick Release seems to work better than many because its altitude pivot point is about two inches forward of the tripod socket, improving the balance with front heavy lenses. One peculiarity of that head is that the azimuth gets stiff and balky in sub-zero weather.

Here are some reviews by SLR Gear
http://slrgear.com/...59/cat/all
http://slrgear.com/...992/cat/13

This is the 70-300mm to avoid
http://slrgear.com/...122/cat/13

Here is Nasim Mansurov's review
http://photographylife.com/reviews/nikon-55-300mm-vr

There's no 55-300mm FX lens. There are several 70-300mm FX lenses, most of which are better because you are only using the sweet spot of the glass so you don't have colour fringing, vignetting, and as much distortion.

But in order to take the most advantage of a telephoto zoom it should have VR, which is only on the more expensive 70-300mm VR lens. It is quite a bit more expensive and heavier than the 55-300mm DX lens, but it is better optically and better built as well as it has faster autofocus with instant manual override. You will be more likely to get better pictures with that lens.

"… Will I get better picture?"

No.

Better picture depends on user skill. The camera and lens, whether they be FX or DX, are only tools. No advantage really.