I grabbed the data from The US Bureau of Labour Statistics page and massaged it into a format that made graphing in a spreadsheet easier. For my spreadsheet, I used LibreOffice, a free and very useful replacement for MS Office. Download it if you'd like to play with my spreadsheets.
So, the statistics that I thought were most susceptible to change if the doomsday predictions were true: Employment, Hours worked, and Terminations (layoffs and firings) ... so here are the graphs (with links to the appropriate spreadsheet).
As you can see, employment tanked after the Bush meltdown, and has climbed ever since. Nary a blip around April 2014 when the first enrolment period ended.
If people were getting fired over Obamacare, we should see a sharp spike in the number of people being fired. Quite to the contrary, the baseline for the number of people being fired jumps during the Bush meltdown, returns to normal around 2011, and then mysteriously drops around 2013 and stays below the 2005 average. If anything, the "Obamacare jump" seems to be a small jump down
"Oh, no, no!" scream the Obamacare haters. "People didn't get fired, they just got their hours cut to part time." So we have to graph the average number of hours worked in a job. (In this case, the BLS stat was the month-to-month change in hours worked, so I just summed this up, and graphed it with the 2005 average being the zero base.)
Once again, we have the Bush Meltdown crater, then things recover (mostly). There seems to be a small drop at the beginning of 2014, but it's more like a blip—then things quickly recover, and actually rise
at the beginning of 2015 to pre-meltdown levels.
So, overall, these stats seem to show that Obamacare has had little, if any effect on employment. If it has had an effect, I would say that it has been very mildly positive. Certainly there are no signs here of the massive doom predicted (and still asserted) by Republicans and other detractors of the Obamacare law.