No, changing your domain name servers (DNS) will not affect your rankings in SERPs. Any change in SERPs is probably coincidental and not in the control of Cloudflare (Google has a fairly complex algorithm for determining placement in search results).
Cloudflare should help your SEO efforts because we help speed up the delivery of content on your websites to your visitors. Please read Matt Cutts' blog on Google incorporating site speed in search rankings. We also provide additional details about Cloudflare and SEO in our blog post.
If you are possibly seeing issues with errors in Google Webmaster Tools that you think may be impacting you in SERPS, please get us the information outlined in this article and open a support ticket.
Cloudflare has a number of optimisations that can be performed to help enhance SEO rankings, see our article on Technical SEO Optimisations for more details.