Do I need a hydraulic jack to do main beam jacking?
The hydraulic jack is an excellent way to be sure you have enough leverage, but more importantly, a high degree of control.  Jackscrews are not as good at lifting by rotating the leadscrew due to high friction, which is overcome with a cheater bar so there is not as much feel developed.  The jackscrew is excellent for holding a load securely, whereas a hydraulic jack might slide if the actuator is not secure.

If you are worried about cost, you can probably get by with a much smaller jack, like 5 or 10 ton.  Harbor Freight has a 20 ton jack for $100.  My local tool rental has them for $8 per day.