You appear to be experiencing the classic symptoms for the issue described here: http://www.intel.com/support/motherboards/desktop/sb/CS-034326.htm. Please follow the instructions in that document.
Oops, sorry. I should have pointed out the last part of that page in particular:
In some cases, the fan speed adjustment does not resolve the issue or can increase fan noise beyond an acceptable volume. To fix the issue, Intel developed a hardware-based solution using a thermal pad. The thermal pad is placed between the mSATA SSD and the bottom of the chassis to improve the thermal performance. To get a Thermal Pad Kit, contact Intel Customer Support.
This issue is primarily seen with the 525 SSDs, so I recommend you try the thermal pad solution next.
Can you get into BIOS using "maintenance mode"? Open the chassis and move the yellow configuration jumper from pins 1-2 to pins 3-4. Then boot the system - it should automatically take you into the BIOS setup.
If that also fails, then may be the NUC needs to be replaced. The quickest way to get a replacement would be through your place of purchase. If that is not possible, you can send an email to Intel Customer Support (http://www.intel.com/support/mailform/desktop/nuc/emailsupport.htm) for further assistance.
When requesting warranty replacement, you will need to include the following information:
Serial number and SA Number (found on a barcode label on the bottom of the chassis)
Please also reference this community thread for the warranty agent to review.