I don't have a clear position on this case, but despite what Catholics say, I think that Mr.Englaro's position was a little more coherent than the one of the Italian government.
From what I understand, the parents' position is based on two amply-condivided principles.
The first principle is that a person has the right to refuse to be submitted to therapy. This principle is (at least implicitly) accepted by the Church. I remember reading some time ago of the case of a pregnant woman who, at risk of her own health, refused therapy that could have harmed the foetus she was carrying, and this behaviour was reported very favourably by the Catholic press. Moreover, when in a few years therapies based on embrional stem cells will become to be offered to patients, I suppose many Catholics will want to be able to refuse them as against their principles. Even Pope John Paul II is rumored to have requested to suspend therapies in the last days of his illness.
The second principle is that when patients are unable to express their will, relatives may testimony the patients' opinion. This is how consent is currently sought for in case of organ trasplant from brain-dead people, for example. This principle is also supported by the Church, that doesn't forbid organ donation.
The only aspect on which perhaps the protesters have a point is whether alimentation and hydration are or not to be considered a therapy. This is debatable, as the means by which Ms. Englaro was kept alive are everything but "natural".
Even with the new law that forces alimentation and hydration not to be interrupted, I don't think a patient in those conditions could survive for long without any other kind of medicament or drug. An immobilized person would develop infections or trombosis without continuos treatment.