Sathi Ho,
 I graduated in May 2011 and started working with a startup company which is e-verified.  I received my new OPT card 2 weeks back  (Under STEM scheme-with 17 months Extension)  & I am getting a nice offer from other company which is directly related to my major & they are ready to sponsor my H1B visa. I do not want to miss this oportunity but this company is not e-verified. I am going to ask the company if they can sign up for e-verification, if they  do it ..excellent..if not my concern is ..is it impossible to work for company which are not e-verifed?
Can you guys please Share your experience if any..
Sabai Ko Bhalo Hos....

Since you are in good status now and have good future/career prospect , my suggestion would be - Dont even invite yourself any problem by working for a company which is not e-verified. Thats the bittertruth.

With the above suggestion, I suppose you should be clear regarding your next move .ie:
Option 1:: request this new cpompany to e-verfiy . Its not a biggie. These companies are afraid of going through such verification since they migt have to present the data/info of other employess too. But at the end of the day its their company and thjeir decision. you can just request them to go for Verification.
Option 2: Work for current company and look for other opportunites. You wil lget it . Just have patience.

~GoodLuck

Thank you Bro..ya I surely will request them if they can signup for E-verification, Since the company I am geeting the offer is a Public Company I am hoping that they will listen it..but any way as u said by end of the day its upto them.. 

Furthermore, You can also mention the following (optional) when you get chance to talk to them:

1. Its not a big deal to E verfiy the company
2. Its a one time shot
3. They need to coomunicate with
4. the verification process is not that long.
5. It will provide more opportunities for the company itself to grow i.e  in terms of  future quality international hires
6. More companies are e-verifying themself these days.

Other helpful source/info for yourself too: http://www.uscis.gov/portal/site/uscis/menuitem.eb1d4c2a3e5b9ac89243c6a7543f6d1a/?vgnextoid=e94888e60a405110VgnVCM1000004718190aRCRD&vgnextchannel=e94888e60a405110VgnVCM1000004718190aRCRD

You  r gr8 bro..I will let you what happens