Upgrading freewheel

[Ajax Bay]

Can I upgrade a 6spd 28 to a 7spd 34 like this one?

Do you have indexed changers? If so, you'll need ones that can cope with 7 sprockets.

People answered your simple question, in essence 'can a 7 speed freewheel screw onto the same hub as a 6 speed freewheel?'
To help the OP find the best solution it would be an advantage to have answers to these questions:
Q1: Why do you want to change the freewheel? Is it to get a lower gear (to climb hills more easily)? Why do you want 7 rather than 6?
Q2: What sort of gear shifters/changers have you got? If they are indexed they'll be 6 speed and will not work with 7 sprockets at the rear (as Tim implied).
Without presuming the Op's answer to those questions, here's a 6 speed freewheel with a 34t large sprocket:
https://www.tweekscycles.com/uk/shimano-6-speed-multiple-freewheel-244794/
My comments ref being satisfied that the RD can cope with both wrap capacity and the 'large sprocket' demands still hold true.

 

[gilespargiter]

The picture of the derailleuer shows a Shimano S.I.S. (Shimano Integrated System) one. I think,(?) the first indexed derailleuer of the types now used. However if it has all the integrated parts it can be set for friction shifting.

I can say from experience - because as mentioned, I use one; that it will achieve a climb up to a 32 tooth sprocket but will not make the massive jump to the bottom gear on the freewheel shown. (I have tried it in the past). It lacks a 'B' screw.
Difficult to tell from the picture but from the relative size of the derailleuer to the wheel, it seems to have cages the same length as mine. It is longer than many road derailleuers but a little shorter than many MTB ones. My poor thing manages chain rings from 52 teeth to 28 and from 11 to 32 on the rear. As you would expect it can't quite take up all that chain, so in the 28 tooth ring the chain rubs itself in the top 3 gears of that range, but basically works fine. It is set on friction shifting, (because I use an 8spd 'cassette' with it) this is much easier to keep adjusted perfectly and I find no disadvantage.

 

[raleighnut]

I'd say it needed a compatible 'Mega Range' derailleur.