Note: If it's been less than 14 days since you purchased the Mini you can take it back for a full refund no questions asked.
At least take it into your nearest Apple Genius Bar and have them get it booted up for you. Be sure to have your Apple ID name and password and your iCloud Account name and password when you take it in.
I was getting the dreaded amber flashing light indicating a firm ware problem and took it in and they had it fixed and me out the door in 45 min. It's a marvelous machine once you get your issues worked out.
Are you trying to migrate your user data from a Time Machine backup? If so take that drive in with your so they can compete the migration for you. Might take more than the 45 minutes however and may be started after you get home.
