I am a US citizen, quit my US job and moved out of US and been living out for about 10 years, can I go back to US and can I claim unemployment till I find a job there.
If you can prove you are a US citizen, you can certainly return to the US. As for unemployment benefits, it will depend on the state, but generally states only provide unemployment benefits based on your recent employment in that state. For example, from Maryland's unemployment FAQ:
An unemployment insurance claim is filed against the state where you worked, not where you live. If all of your work in the last 18 months has been in a state other than Maryland, your claim should be filed against that state. The laws of the state you are filing against govern your claim. In most cases, you will be instructed to contact the state where you worked directly