The reason amino acids exist largely in their zwitterioniczwitterionic form at biological pH $\mathrm{pH}$ (usually around 7) is due to the pKa$\mathrm{p}K_\mathrm{a}$ of the constituent groups. It's not that the oxygen 'wants' to lose a proton, but more that at that pH$\mathrm{pH}$ the equilibrium lies towards the deprotonated state (things are rarely 100% protonated/deprotonated).
In general, carboxylic acids easily lose their proton to form carboxylates (the anionic charge is delocalised over the three atoms making the proton loss 'favourable'), and amines act as bases, to pick up a proton.
At a pH$\mathrm{pH}$ of 7, the medium is basic enough to cause the carboxylic acid to be deprotonated, and the amine to be protonated. If you look at the diagram below, you can see that increasing or decreasing the pecan shift the equilibrium towards fully protonated (at low pH$\mathrm{pH}$) or fully deprotonated (at high pH$\mathrm{pH}$) amino acid.
Ionisation state of amino acids over 0-14 pH$\mathrm{pH}$ range, taken from Biochemistry, Berg (7 ed.)