Ask yourself: if you spent all day swatting mosquitoes one or even a few at a time--and even if you were GREAT at hunting them down--would your yard suddenly be free of mosquitoes? That's doubtful. Mosquitoes will just fly in from your neighbor's yard. They don't know a whole lot about property lines. If your neighbor has a wet spot where they breed--even just puddle beneath a wood pile--mosquitoes reproduce in such numbers that there will be way more skeeters than your chickens can control by foraging. To control mosquitoes in your area, you'll want to eliminate standing water in your yard. And in your neighbor's yards. And in any public areas nearby. And if you can drop the temperature to freezing, that would help, too. The problem is that mosquitoes don't need a pond or swamp to multiply. They'll reproduce in water standing in potted plants, or a rain gutter...or even just an old bottle cap lying unseen under your crawlspace or porch. And the whole life cycle can take as little as four days.
Ticks may be a slightly different story. They're not especially fast movers, and aren't as prolific as mosquitoes. A few studies have demonstrated that chickens are effective at reducing the number of ticks in a particular area. Under the studies' conditions, chickens showed they are capable of eating hundreds of ticks in a very short period of time:
Influence of exposure period and management methods on the effectiveness of chickens as predators of ticks infesting cattle
Predation of livestock ticks by chickens as a tick-control method in a resource-poor urban environment
Predation on livestock ticks by chickens
Just how you'll experience that impact of reduced ticks from your chickens is questionable, though.
While we have people who report anecdotally that their dogs get no ticks now that the chickens are around, we can assure you that there are plenty of chicken owners whose dogs still suffer from ticks. To control ticks, like controlling mosquitoes, you'll want to eliminate their habitats. That means avoid tall grasses and no build-up of leaf litter or other detritus in the landscaping. This is where the chickens come in to help: these are areas where they love to forage.
They're not so good at foraging swamps (or potted plants) for mosquitoes, but they will patrol the perimeter of your yard looking for tasty, many-legged snacks. They will toss leaf litter like confetti (and your mulch, if you don't take the proper steps), and they will forage in any long grass and areas you leave untrimmed. (But weed whacking is one of the joys of your life, so you don't have any areas like that, right?)
If you live in suburbia or--er, urbia, having chickens in your yard will probably mean you will see few, if any ticks... presuming your neighbors don't provide lots of habitat. And if you live rurally, you might see fewer in your immediate area. But you can't exactly eliminate them because their habitat is everywhere. You can't remove all the leaf litter from a forest. And most people don't want to go to the time and effort of mowing acres and acres of pasture so there are no long grasses. You could always put in some parking lots--if you're crazy--but the experience of being surrounded by concrete is probably not why you moved to the country in the first place!
The bottom line is, while we can't guarantee chickens will eliminate every mosquito and tick from your yard, they will certainly do their part, and as part of an integrated pest management strategy, you could see vast improvement in the reduction of these crawling pests around your home.