I got TinyGPS (v12) to work with DUE. GPS EM-406A is 5v, but spec and scope show TX and pps output are only 2.85v, so no need for level converters. For IDE 1.5 to see the TinyGPS library I had to rename Examples/ to examples/. All three examples worked, replaced SoftwareSerial with Serial1, and for static_test sketch I had to neuter the PROGMEM stuff.
You don't even need the TinyGPS lib to test the pulse-per-second output (1 microsecond accuracy). I hooked the pps output to a DUE digital port and used attachInterrupt() (required IDE 1.5.1) witht a micros() timestamp, and used that to measure the clock frequency of the DUE. The simple gpspps sketch and results can be viewed at https://github.com/manitou48/crystals