[SalesForce] ApexPages.currentPage().getParameters with javascript remoting

There are a couple of things.

1. In Apex, init() methods are not called by default. Any code you want to run during the instantiation of an Object must go in the constructor, see Using Constructors. Since JavaScript Remoting requires your methods to be Static, setting Instance Variables will not do you any good.
2. Your static method, "findAccounts" is trying to reference an instance variable, "accounts". I got a compile error when initially trying to save your class. If you want to reference the "accounts" variable from within another static method, make sure to define this as a static list.
3. Since init() is never getting called, the list of accounts will be null. If you call this init() method within the findAccounts method, then the data will be populated. Note, that you must change init() to be a static method then, as well. As referenced above, it's generally not a best practice to query all objects into memory, as you will soon hit governor limits for the total number of records retrieved by SOQL queries of 50k query rows.

If you put all that together, this should get you at least a working example.

global class MyClass
{
    public static List <Account> accounts = new List <Account>();

    public static void init()
    {
        accounts = [SELECT Id, Name FROM Account]; 
    }

    @RemoteAction
    global static Account findAccount(String accountName)
    {
        init();
        for(Account acc : accounts)
        {
             if(acc.Name == accountName)
                 return acc;
        }

        return new Account();
    }
}

Note, that this will infact requery the database everytime findAccounts is called. A design pattern I like to use, and have seen a lot looks something like the following:

public static List <Account> accounts;
public static void setAccounts()
{
    if (accounts == null){
        accounts = new List<Account>();
        accounts = [SELECT Id, Name FROM Account]; 
    }
}

This will ensure that the query is only run once within the lifespan of the request.

You also may want to utilize a Map<String, Account> accountNameToAccountMap see Apex Maps, so that you can avoid iterating through the list every single time. But, I'll leave that as an exercise.

Make sure the apex page has your controller set as the page controller?