1. She pushed me to accept her order.
- Grammatically correct. It means she tried to persuade me to accept the
order. Out of context, we don't know whether i accepted it or not.
2. She pushed me into accepting her order.
- Grammatically correct. Out of context this implies that I did accept
the order.
3. You can count on him to help you.
- Grammatically correct. It means you can be sure he will
help you.
4. You can count on him helping you.
- It's not clear (exactly) what this means as a standalone sentence out
of context. The Ngram
finds no example, though some people do say this with the -ing
form with some other verbs. The standard construction is „count/rely/depend on
someone to do something“.
(Ngrams not found: count on * helping)
Here's an example:
At just about every one of my meetings, I can count on someone saying the
official inflation rate is useless because it excludes prices of two of our most
important necessities, food and fuel. (cals.nscu.edu)
This sounds idiomatic and perfectly acceptable. With „say“, the Ngram
does find examples with the -ing form, but fewer than with the
to + INF form.