Does the Higgs boson give mass to ALL other particles? The Higgs Field gave mass to other particles via spontaneous symmetry breaking; does this mean it gives mass to all particles that have mass - such as neutrinos, quarks or weak bosons and strong bosons (gluons)?
In the articles I've seen its not clear what the criteria is for the Higgs Field to couple to particles to give mass; Anna v's answer suggest that electo-weak vertexes are required; but the linked article suggests it also couples to quarks.
 A: Gluons are bosons, they have spin one and are mediators of the strong force. They have mass 0 and there  is no weak interaction vertex with a gluon.
The extract is talking of the Higgs field, not the Higgs boson. A field in physics is 

A field is a physical quantity that has a value for each point in space and time  at that point. 

The Higgs field permeates all space and is distinct from the Higgs boson. It is the field that gives the mass to the Z and W when the electroweak symmetry is broken. Symmetry is broken at every point is space  and time ( the mexican hat ) that describes the field at each point. 
The Higgs boson is a particle that appears because of the group structure of the Standard Model, and is on par with all the other particles. 
It is the Higgs field that gives mass to the particles  of the theory which have electroweak vertices ( interactions) and to the Higgs boson itself. .
edit after comments.
Note that the Higgs field gives masses to elementary particles. If you see that they are massive in the table it means they have weak interaction vertices and the Higgs breaking of symmetry raises them from the zero mass they would have had  before breaking.

These are the particles that build up all matter
The electrons appear in the atoms, the quarks are always bound and appear according to the rules of SU(3) color. The three quarks that compose a proton, for example , exist in a tight "bag" exchanging gluons with each other, and the gluons exchange gluons with each other etc. in a "soup" of contained interactions that can be computed  or approximated. This means that there are binding energies and invariant masses between all these constituents that give the mass we measure for the proton, while the central three quarks have very small masses as you can see in the table ( or go to the link for enlargement). This is due to special relativity that reigns at these dimensions and energies.
The quarks do have electroweak vertices in interacting with leptons , a simple example neutron decay:

A: The Higgs field is not giving mass to all other particles. There are particles that acquire mass differently - for example neutrinos. Or the yet undiscovered particles of dark matter probably don't get mass from the Higgs field. Also please notice the difference between the Higgs field and the Higgs boson. The Higgs field is giving mass to some particles, but the Higgs boson is just an excitation of the Higgs field and is not giving mass to anything.
The fermions acquire mass when the Higgs field couples the left and right chiral massless fermions into the massive Dirac fermions.
