Do all mosquitoes spread Dengue Fever or Chikungunya?

No. Aedes aegypti is the species of mosquito which is primarily responsible for spreading dengue. Also, only female mosquitoes can spread dengue or chikungunya.  Males do not bite or spread disease (in fact males cannot bite). Other Aedes species such as Aedes albopictus, can also spread dengue and chikungunya, but Aedes aegypti is the main vector and it is this species that is responsible for almost all of the dengue epidemics. Aedes albopictus is often cited as a dengue vector and can indeed spread the disease but it is not very efficient. Other mosquito species bite humans but do not spread dengue fever.

Further reading:

Jansen, C. C. and N. W. Beebe (2010). The dengue vector Aedes aegypti: what comes next? Microbes Infect. 12(4): 272-279.

Lambrechts LScott TWGubler DJ.(2010) Consequences of the expanding global distribution of Aedes albopictus for dengue virus transmission. PLoS Negl Trop Dis. 2010 May 25;4(5):e646..