Current source with IGBT
No. Not as shown without "select on test"ing.  

For input purposes an IGBT looks like a MOSFET. It will have a minimum gate voltage at which conduction begins and this will vary with the model of component and across a significant range for a given model.

For example here's the data sheet dor a FGD2N40L IGBT  - a baby one by most standards. TO252 package, 400V, 7A continuous. On page 2 it lists Vge(th) - the turn on voltage as 0.70V minimum, 0.85 V typical, 1.2V maximum. If you used this part in your circuit the "constant current" for a range of devices would be significantly variable unless you adjusted them for each circuit used. Use of a large voltage across Re would minimise error by "swamping" the variation in Vge(th). 

However, if you monitor \$V_{RE}\$ with an opamp or similar and use it to hold \$V_{RE}\$ constant by  driving the IGBT as requird, you could have an acceptably accurate current source.