I have a missions trip planned for this summer and will fly to Anchorage, AK, with a group but do not want to fly home. I was thinking about taking the train back through Canada. I've had an absolutely horrendous experience with Amtrak and will not give them another dime after standing in sub-zero weather for five hours while their waiting station was locked up tight and then the engine broke down countless times.
Can anyone recommend a company to help me organize the trip. Is it a U.S. company or a Canadian company that would pick me up in Anchorage and drop me off in Chicago?
Or possibly, I might bus from Anchorage into Canada, Via Rail across half of Canada, and then bus again from somewhere in Canada back Chicago? Time isn't an issue.

There is no train route between Alaska and the lower 48, and no scheduled bus service from Anchorage to anywhere in Canada.

Your options are to fly, or take a series of ferries from Whittier to Juneau to Bellingham, then to Vancouver or Seattle, or buy (not rent) a car and drive, or take a cruise ship from Whittier or Seward to Vancouver and a series of trains from there.

I am sorry that you had such a bad experience on Amtrak.
I take it that you were on the Empire Builder. Where was your "waiting station"?
Fly from Anchorage to either SLC or Denver and give the California Zephyr a try.
When Amtrak engines break down, they are sometimes pulled by BNSF engines.
If you call Amtrak (1 800 USARAIL) and explain the train you were on and the date, they MAY give you a voucher for some free travel.

<<Fly from Anchorage to either SLC or Denver >>

The OP clearly stated "I do not want to fly home" and refuses to use Amtrak.

Without flying or using Amtrak this will be a long trip with many changes of transportation. Alaska Ferries to Prince Rupert BC, then Via Rail Trains or bus Prince Rupert to Jasper AB, Jasper AB to Toronto ON, Toronto to Windsor ON then bus from Windsor to Chicago, or bus from Toronto to Chicago. I suppose bus from Winnipeg MB to Chicago is a possibility as well.

Gardyloo's suggestion would work as well, but would also make for a long and involved trip.

Take a look at www.rometorio.com and use the public transit button. I don’t expect you will find anything much there that hasn’t already been mentioned, but it should at least show you how limited your options are.

The cross country Via Rail sleeper class is a wonderful way to cross the country. My recommendation would be to flying from Anchorage to Vancouver, then Via rail to Toronto which ought to give you some options, including through niagara falls and Buffalo. Or you could continue on by Via to Windsor, pop across the border to Detroit and make your way from there. I don’t have any experience or knowledge of west coast ferries, but if you could make it work, the Via cross country train train from Jasper would be a good option.

Between Anchorage and Chicago you will have to travel through vast areas which are sparsely populated or uninhabited. (And of course those great lakes do get in the way. It is understandable but very frustrating that there is so little public transportation.

take a one way cruise back to vancouver. www.vacationstogo dot com.

then trans canada train across most of canada. then buses?

sorry.. had not read gardyloo's suggestion.. but I think it is the logical one