(Please take this as constructive, as I very much want to see the global eradication of biting mosquitoes occur.)
I think this specific proposal (an online petition/Facebook activism) is naive and likely counter-productive. I feel like I should be docked several thousand Initiative Points for saying this, but please don’t do as you propose.
For starters, you cannot say “mosquitoes”—as others have pointed out, there are ~3500 separate mosquito species, only ~100 bite humans, and only several dozen transmit disease. Narrowness is a virtue here, and this level of biological imprecision could alienate potential allies who will take you as reckless and uninformed.
(A related point is that the most promising interventions for eradication (like the sterile insect technique) are species specific, so it makes sense to start with the highest-priority target. Because [complex chain of reasoning to fill in later], I think aedes albopictus is likely the best bet.)
Also, I don’t think country-level eradication plans (even for a single species) have the slightest chance of working long-term due to persistent re-invasion risk. A continent- or hemisphere-scale plan would be required, which comes with the commensurate coordination problems, and is much less likely to be aided by petition.
Additionally, don’t underestimate the potential for politicization of such a program. Raising it to the level of public awareness without a good communication plan is premature.
Finally, many folks have entirely reasonable concerns about downstream effects that really do deserve sober analysis. I think it’s likely that effects on other species or ecosystem stability will be negligible (or at least worth the cost), but that’s an empirical question that deserves serious attention. As someone else pointed out, this is probably the key objection to overcome, so you might want to invest some effort in alleviating it upfront.
(All that said, it’s awesome that you’re thinking about this seriously. The eradication proposal is sort of my favourite idea ever, so please PM me if you’d like to discuss it further offline.)
Sleeping sickness is transmitted by the Tsetse fly, which is not a mosquito. Even ignoring this I’m unsure what the effect on sleeping sickness has to do with environmental impact—this is the target effect of the program, no?