Don't know about this particular charter but Miami Air have been over here before with their 727s, could be them.
Regarding this type of trip there are very few options available other than 727s (and a few 737s and even fewer 757s). No modern airliners are available as they have to fly round the clock to justify their existence, especially mid-summer so cannot be spared for long enough. This leaves a niche for operators who can make a profit using written down ex-airline aircraft (with nice new interiors) with relatively low utilisation. There are also one or two 757s doing this sort of operation (Titan in the winter and also one in Spain) but other than that there is not much choice.
The latest Boeing BBJs and A319CJs tend to be laid out with too few seats and of course cost a packet.