A friend was on a diet and was told that if he does not want to eat bread on Shabbos, he does not have to. Is this correct? Do we have an obligation to wash for bread and have three meals on Shabbos?
There is a requirement to eat bread for first two meals of Shabbos, as per the Shulchan Aruch Orach Chaim Siman 274:4.
However, the Mishna Berura Siman 291:1 Sif Katan 3 notes that Shabbos meals were given for oneg meaning pleasure, not for tzar meaning pain or unpleasantness.
The sefer Shemiras Shabbos K'Hilchasa therefore rules in chapter 54 paragraph 35 that someone who is concerned that Shabbos food even may damage him is exempt for eating that food. This of course includes bread, as the entire chapter there is dealing with the requirement to eat bread on Shabbos.
For further reading, see the footnotes there in Shabbos K'Hilchasa where he brings the Mishna Berura Siman 288:1 Sif Katan 3 that eating food which you know will damage you may even be forbidden.
This site is not in exchange for Rabbinic guidance. This answers the second question as to whether there is an obligation to wash for bread on Shabbos, which according to the Shulchan Aruch there clearly is. There may be instances where a Rabbi would give a dispensation for a particular circumstance, which may be the case here. Yet in a regular case one would be required to wash.
If for health reasons you cannot eat bread, it's OK. If eating bread will harm your health or make you sick, OK. But if you are just "dieting," there is no exemption for that - and, by the way, all you have to eat is one or two slices of bread, which will not hurt your diet.
The inyan of shabbos is oneg. If eating bread for some reason causes one to lose their degree of enjoyment, then they shouldn't. The Cohanim had reasons to eat offered portions of grain, bread, meat, wine, oil, etc. The people of Israel have no particular requirement of what is to be eaten on Shabbos other than what brings them joy. The shoresh of joy, simcha, is the same as Moshiach, shin-mem-chet. Moshiach is the one who restores the joy of the people.