(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.)
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.
I don't see any reason to only target those that transmit diseases. Target ones that are simply annoying because they string the average person, gives everyone a clear reason to support the proposal. There are also people with allergies or who simply don't heal the stinged area very well.
...Also, I don't think country-level eradication plans (even for a single species)
According to Louie Helm eradicating a species of mosquitoes could be done for as little as a few million dollar.
I don't have a few million dollar lying around so I can't spend my own money to do it. On the other hand, I think that on average every German citizen would be quite willing to pay 1€ per year to rid Germany of mosquitoes that bite humans.
That means it's a problem of public action. The German government should spend 80 million Euro to rid Germany of Mosquitos. That's an order of magnitude higher than the numbers quoted by Louie Helm).
The same goes basically for every country or state with mosquitos.
How could we get a government to do this without spending too much money ourselves? The straight forward way is writing a petition. We could host a website and simultaneously post a petition to every relevant parliament on earth.
How do we get attention for the petition? Facebook. People don't like Mosquitos and should be willing to file an internet petition to get rid of them. I would believe this to spread virally. The idea seems interesting enough to get journalists to write articles about it.
Bonus points:
After we have eradicated human biting mosquitoes from our homelands it's quite straightforward to export the technology to Africa.
Does anyone see any issues with that plan?