I know nothing about the "black balling" method mentioned in the problem, but hearing the term got me thinking about an easy solution:
Put the same amount of quarters in a bag as there are participants. Pass the bag around and if your answer is yes then take one quarter out. If your answer is no then put a random number of quarters in a bag - the key here is you put as many quarters as you want in there. At the end of the vote if the bag is empty - it's a unanimous yes. If there are quarters in a bag, someone voted no, but you won't be able to tell how many no votes or yes votes there were, if any.
The only issue I see with this system is:
If there are less quarters in the bag at the end than there were participants to begin with. This would undoubtably mean that someone voted yes. However, you cannot make a rule that if you vote no you need to put at least the number of participants worth of quarters in there as that would mean that if the bag contains exactly the minimum then it would be obvious that only 1 person voted no. So, being the savvy participants that they are, the rule is unwritten and unspoken, but if you vote no you make sure your random number of quarters you put in there is greater that there were people to begin with. At the end of the vote nobody should be able to tell how many people voted no.
Edited to add:
Some people may take issue that the last person to vote will be able to tell there is only one coin left in the bag, so if he was voting no then he would know that every one else voted yes. To remedy this the bag could start with the number of participants worth of coins plus 20 coins, say; that way the last person will have a bag full of coins, short of counting them all while voting he won't know the exact amount. To make it hard to cheat you can put 100 extra coins in there, that would take a considerable amount of time to count.
Also if there was only 1 person voting no, then if the final count is revealed the person will be able to tell if he was the sole no vote or not. To remedy this the final count should not be revealed, if there is even a single extra coin in the bag at the end of the vote then the matter should be decided. After the vote is decided, the bag should then be passed around the participants once again and this time any random amount of coins from each participant should be thrown in or taken out. At the end of this procedure not even the sole no vote will be able to tell that he was the only one.