Q:

How can I export or query all hospitals from Open Street Map?

I'd like to extract all the hospital locations in the US from Open Street Map as a reference for my geocoding project. I know how to get all the OSM data for a small area with the QGIS plugin but not sure how to query a larger area?

A:

Two more ideas:
1) The read-only Overpass API.  I don't know that it'll work for the whole US in one pass, but maybe if you extend the time out enough...?  For a smaller area (.5 degree max) and with the benefit of a (minimal) UI, you can access Overpass via the XAPI Query Builder.  There, you can put amenity=hospital in the tag search, select your area, and go.
2) Geofabrik downloads, filtered with Osmosis, as described in How to extract partial data for large regions? on the OpenStreetMap Help.

A:

Nominatim can do this but on a large scale it would be more efficient to download the USA extracts and use your own.
Nominatim Example:
http://nominatim.openstreetmap.org/search?q=usa[hospitals]&format=json&polygon=0&addressdetails=1
Returns:
[

    {
        "place_id": "640932",
        "licence": "Data © OpenStreetMap contributors, ODbL 1.0. http://www.openstreetmap.org/copyright",
        "osm_type": "node",
        "osm_id": "257489577",
        "boundingbox": [
            "42.3371238708496",
            "42.3371276855469",
            "-71.1060409545898",
            "-71.1060333251953"
        ],
        "lat": "42.3371263",
        "lon": "-71.1060391",
        "display_name": "Children's Hospital, Blackfan Street, Fenway, Boston, Suffolk County, Massachusetts, 02215, United States of America",
        "class": "amenity",
        "type": "hospital",
        "importance": 0.44314331355821,
        "icon": "http://nominatim.openstreetmap.org/images/mapicons/health_hospital.p.20.png",
        "address": {
            "hospital": "Children's Hospital",
            "road": "Blackfan Street",
            "suburb": "Fenway",
            "city": "Boston",
            "county": "Suffolk County",
            "state": "Massachusetts",
            "postcode": "02215",
            "country": "United States of America",
            "country_code": "us"
        }

for xml replace json with xml
Source:
http://wiki.openstreetmap.org/wiki/Nominatim
You might be better off asking here:
https://help.openstreetmap.org/

