The reason it doesn't work, when you pass a symbol that refers to a var that holds a map, is because the macro is only seeing the symbol literal — not the value it refers to. When you pass it a map literal, the macro is seeing the map literal.
If you wanted it to also support symbols, you'll need to resolve the var the symbol refers to and get its value e.g. (var-get (resolve bindings)):
(defmacro block [bindings & body]
`(let [~@(mapcat (fn [[k v]] [(if (symbol? k)
k
(symbol (name k))) `~v])
(cond
(map? bindings) bindings
(symbol? bindings) (var-get (resolve bindings))
:else (throw (Exception. "bindings must be map or symbol"))))]
~@body))
(block ctx (if true (prn "true" a b) (prn "false")))
"true" 3 4
(block {:a 3 :b 4} (if true (prn "true" a b) (prn "false")))
"true" 3 4
You also need to "splice" body into the form using ~@, because body will be a sequence of forms even if it only contains one form.
It can also help to look at how your macro expands when troubleshooting it:
(macroexpand-1 '(block ctx (if true (prn "true" a b) (prn "false"))))
=> (clojure.core/let [a 3 b 4] (if true (prn "true" a b) (prn "false")))
