Will a boot provide more support than a shoe?

You cannot assume that a boot is stronger than a shoe - you still need to check the heel counter for the actual degree of stiffness; but often boots do have a heel stiffener applied to a greater height than a shoe and in this case it will provide more support.  But this is only the situation when the boot conforms to the shape of the ankle.  When the boot is on, see if you can get your fingers down the sides below the ankle bones.  If the gap is sufficient for your finger to go easily down and you can pull the boot away from your foot, then the boot will not be supportive and a close fitting shoe may be better.  

Boots still need to have an adjustable fastening mechanism at the front to tighten them to the foot and keep the heel back in the heel counter.  A zip is not sufficient.  However, if the boot has laces that can be adjusted initially, the zip fastening can be used once the initial tightening is achieved.

What make shoe or boot would you recommend?

Because everyone has different shaped feet, no one brand of shoe or boot will suit everyone.  My recommendation is generally to go to a shop where the fitter is experienced so they can choose the footwear to suit the foot.  A shoe shop that sells a range of brands is recommended so that the fitter can measure your foot and then has a wide range to choose from.  Each shoe manufacturer uses a slightly different last shape and therefore some shoe brands are good for high insteps, others are good for wider feet, others for narrower heels.  This is usually based on native population in the country of manufacture.  Find a shop that sells a range of brands, for example, a shop selling makes such as Ricosta, Angulus, ECCO, Geox, Garvalin, Petasil, Startrite, Clarks.  

If your child has problematic feet or uses orthoses, you may need to phone in advance and ask when the most experienced fitter is working so you can time your visit to their working day.  When orthoses are used, the experienced fitter is often very good at choosing a shoe / boot that has a removable inlay so that more room can be made for the prescribed orthoses.  Sometimes only the back part of the inlay needs to be removed so the front section is cut off and replaced.  For this expertise and knowledge of shoe fitting, you should expect to pay more for the shoe you are buying.  If you buy a pair of shoes made by a recognised shoe manufacturer (and not a designer label brand) you will be receiving a better quality product and these are more costly that those sold by the High Street fashion stores.

Avoid buying a shoe that has a prominent rim where the sole is wider than the upper.  This often causes increased tripping in the less stable child.